Do Le et al. (2012) train all three autoencoder layers at a time, or just one?
The paper refers to layers and sub-layers, and clearly indicates that one layer includes all three sub-layers, so when they say they train all three layers simultaneously, they are talking about the three autoencoder layers, not the sub-layers.

This also agrees with the fact that only the autoencoder layer has tunable parameters.  The other two layers use uniform weights.

I do not have a good answer for the follow-on question yet... would love to hear what others think!